Lagrangian Cobordisms, Dehntwists and Real Algebraic Geometry
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Formal Metadata
Title 
Lagrangian Cobordisms, Dehntwists and Real Algebraic Geometry

Title of Series  
Number of Parts 
36

Author 

License 
CC Attribution 3.0 Unported:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. 
DOI  
Publisher 
Institut des Hautes Études Scientifiques (IHÉS)

Release Date 
2015

Language 
English

Content Metadata
Subject Area  
Abstract 
We will explain the relevance of Lagrangian cobordisms in Lefschetz fibrations to the study of the (derived) Fukaya category of the fiber. In particular we will give a cobordism interpretation of Seidel's long exact sequence, introduce cobordism groups and also outline how to study real algebraic structures using cobordisms. The talk is based on joint work with Octav Cornea.

00:00
Point (geometry)
Color confinement
Observational study
Multiplication sign
Distribution (mathematics)
Manifold
Number
Plane (geometry)
Lecture/Conference
Subtraction
Position operator
Fiber (mathematics)
Condition number
Focus (optics)
Theory of relativity
Product (category theory)
Algebraic structure
Infinity
Counting
Critical point (thermodynamics)
3 (number)
Set (mathematics)
Connected space
Category of being
Mathematical singularity
Right angle
Object (grammar)
06:41
Point (geometry)
Greatest element
Multiplication sign
Spiral
1 (number)
Manifold
Canonical ensemble
Event horizon
Exclusive or
Positional notation
Thermodynamisches System
Lecture/Conference
Fiber (mathematics)
Focus (optics)
Spacetime
Rational number
Chemical equation
Projective plane
Physicalism
Algebraic structure
Critical point (thermodynamics)
Line (geometry)
Sphere
Functional (mathematics)
Connected space
Category of being
Fiber bundle
Object (grammar)
Identical particles
Family
Freezing
13:31
Divisor
Multiplication sign
Direction (geometry)
Sheaf (mathematics)
Mereology
HurewiczFaserung
Event horizon
Frequency
Lecture/Conference
Square number
Length
Absolute value
Fiber (mathematics)
Units of measurement
Process (computing)
Physical law
Neighbourhood (graph theory)
Moment (mathematics)
Volume (thermodynamics)
Critical point (thermodynamics)
Sphere
Functional (mathematics)
Geodesic
Field extension
Formal power series
Symmetry (physics)
Network topology
Mathematical singularity
20:34
Multiplication sign
Projective plane
Infinity
Coordinate system
Game theory
Mathematical model
Bilinear form
Fiber (mathematics)
Number
23:17
Point (geometry)
Axiom of choice
Complex (psychology)
Group action
Latin square
Direction (geometry)
Manifold
1 (number)
Survival analysis
Student's ttest
Likelihood function
Proper map
Frequency
Inclusion map
Plane (geometry)
Thermodynamisches System
Lecture/Conference
Ranking
Subtraction
Position operator
Fiber (mathematics)
Units of measurement
Condition number
Social class
Area
Theory of relativity
Spacetime
Mass flow rate
Element (mathematics)
Mathematical analysis
Infinity
Algebraic structure
Mathematical model
Mortality rate
Price index
Set (mathematics)
Perturbation theory
Ring (mathematics)
Limit (category theory)
Measurement
Substitute good
Database normalization
Right angle
Fiber bundle
Object (grammar)
Untermannigfaltigkeit
Marginal distribution
Resultant
38:47
Point (geometry)
Computer programming
Group action
Multiplication sign
HurewiczFaserung
Explosion
Inclusion map
Lecture/Conference
Area
Product (category theory)
Spacetime
Curve
Algebraic structure
Independence (probability theory)
Critical point (thermodynamics)
Bilinear form
Local Group
Hand fan
Vector space
Mathematical singularity
Normal (geometry)
Right angle
Object (grammar)
Family
Resultant
47:14
Axiom of choice
Complex (psychology)
Building
Group action
Multiplication sign
Sheaf (mathematics)
Water vapor
Inverse element
Complete metric space
Mereology
Explosion
Maxima and minima
Positional notation
Physical system
Spacetime
Theory of relativity
Mass flow rate
Generating set of a group
Complex manifold
Staff (military)
Hyperbolische Gruppe
Price index
Hausdorff dimension
Order (biology)
Mathematical singularity
Right angle
Resultant
Point (geometry)
Classical physics
Observational study
Link (knot theory)
Civil engineering
Connectivity (graph theory)
Real number
Similarity (geometry)
Student's ttest
Distance
Theory
Number
Frequency
Goodness of fit
Causality
Thermodynamisches System
Lecture/Conference
Natural number
Energy level
Subtraction
Series (mathematics)
Focus (optics)
Chemical equation
Projective plane
Bound state
Algebraic structure
Division (mathematics)
Ring (mathematics)
Set (mathematics)
Mathematical model
Sphere
Logic
Network topology
Iteration
Object (grammar)
Game theory
Family
Local ring
00:03
around thanks for the invitation from Seoul so this is joint it's based on joint work with the stuff going on and so I was told that they should also try to mold moreover the definitions and the motivation for the subject I'll try to combine would thinks so the undermining the main the object of study here comes along rise in accordance with the elections whom Gration so I should probably explains why the women weightlifters who so there are many many the new answers in different definitions with the 1 I will adopt here use coalescence Impressionist has about those faces he and comes with a map to the complex plane in my case and this thing is that those who is plenty of structures like the syntactic structure on the there is an almost public splattering and my little Ireland in the complex structure of the plane and I will assume that by his count the Norfolk respectively structures cost this Jaisal compatible with missile made up there earlier Of that isolated critical points isolated critical points in the whole of all the lonely numbers .period the only be so this means that these JayZ's venerable near the singularity and nearly singularity and right confines of Norfolk charts making by really looks like a more freak most function and is about ordinary enough so and I will not put an additional assumptions which is the traditionally of it is what it is it's called pain so in his outside of town outside of all things that you so nobly said I'm going to consider is basically something like mad but you that's the place so I could the outside the few he further if you can't trivialize everything all title you sold the assumption is that the restricted stock the continent of fuel viewed as in manifold he's really isomorphic to a product of all the seemingly endless you I was a significant for M is handled with the splits syntactic structure also is going to be a cornerstone buying the standouts include the structure of the plane barracks sound with some syntactic structure in OK and sees positive course and downwind we because the generic fiber generic my focus so this looks like a very Strong condition however every every every religious relations can be adjusted to so so every lacerations Yukon I don't want to go to the precise with every necessary so you can keep it the way it was you know In some you like this and that Infiniti change the structure such that we don't think this is you have the shortest from it should it's a U shape each year and that his mother in you fired away from the point of you have to have a lot of the Yale and while I didn't understand the properties but the Yates is supposed to deal with the marshal it was the 2nd time that each of the proper nonaligned in that many conditions here and at the end of going into 1 yeah I assume that there is a compact however I am I will I'm allowed to assume that there is no convicts at infinity too but then I have to assume that if you feel the hardest thing without throwing you History alive so we have to like spent half an hour on the precise definitions but some the message I want to convey independent of those things anyway so it's possible the bus tour to the setting and the important thing about this repressions is that they come together with a collection there's a connection and the connection is just well below the horizontal distribution will always be in all only only then of the complement to the fiber the convention doesn't exist everywhere that exist only away from the political point and so it exists connection on the plight of the critical point In this means you can do about plans for the 3rd day year of not only endangers users so did apologize so once again only
06:41
the fibers Nelson eclectic anything so only get Amanda although complement and this gives you something to the mention of the project as more frequently please OK so why fuss about things so I wanted very convenient ways to identify the fibers away from the Ushaped and because of trivialization the when affixed to the connection is chairman of the the Bottoms bonds balls In the continent of fuel is trivial and identity and so this is very convenient for this these and other objects that exist here of the family the invention cyclists vanishing spheres so again I'm going to do this it was very rigorously enforced so basically if you take a critical point as a critical value it said that the political that's the only the critical value while you take a bath emanating from it the so on this structures together but give you a subcommittee 74 in the Goma before the money and so this is the English along with them in this thing has the following properties so the government is in place is pulled even in the Punjab what fails in these environments the is good value on the college sports proposed ballot unschooled longer and I don't think so how does it look into it it was like this his all that we could basically the news over every point you know don't you get the sphere inside the fight over that .period except of course over the .period reserves well you get the point so what analysts think looks looks like a copy of this letter this land lines inside the space and this is fears here of the vanishing of diminishing spheres of because they just go shrink and shrink and vanished from his office as you will years so yeah so technicians ready for said Z's which is loaded 0 p dumber intersect the fiber over is is sphere London here In the end user and the intersection uses 0 is just 1 point I hope you know it's not but it is hard to construct this with the Bells canonical construction altogether and 1 way for example is 1 of the most elegant ways to take a look at modern rival Norfolk most functions I just the syntactic structure to be competitive with gluten constructed locally and then moved by politicians pulled down in approved the millenium construction for for the fingers and again this week they watched I'm too willing to another thing that will play a role here you where land ones used you specify gone Was it specifying the it if the the it the event was the 1st time yes when 1 of the world's injuries focus well and in what makes it unique is the fact that it's in my hand on the ball on to flee dropped the reference to the fire look but that it was not clear solid ,comma focus well now the most important was another important thing is the dentists Peter Lindquist the benefits for the event with a message or a thinkers a nothing to do with the delicious food rations so you love syntactic manifold AEK's going up and you have a sphere decided the spheres should be if you want to be represents should be provided rise the mothers feel so it's really of meant assent to and that is the on we have an abundance here the London that thing the then all of these things in here it used in basically canonical way O'Donnell physical syntactical freezing cold towels so style the let's that's that's right here and to be the image an and of course and doesn't know how it will spiral drives sold as an abuse of notation here but that was the right style and the so balances of an upfront
13:22
exclusions and lets us in the office the and it is supported the
13:32
year come in How do you basically constructed so what you well you hold and comes always was a leader but by time than neighborhood went inside thinks so little enables affair looks like the abandoned 1 all of fear inside its sold the event talking about the combatants apart and the new council we can construct then installed inside the so we are going to acknowledge genetic flaw From the unit the abandonment of office and my Lucille section to itself and this is what I would call the normalized normalized To those job so a lot normalize those lands only the direction of the victim the length doesn't and therefore zeal section you don't get it so attractive a common Bonnie if you take the hunted and functional of killed to be the square With along India's by the eventually you get exactly at this map In and what is that this kind of .period on what is going is important here is that see fine by and that's very special for the sphere extent extends the little section the little section and becomes a little sexual exactly the people them on this section of the people that this and now basically want to exploit this special symmetry of the sphere and extend everything so you want to start 0 restrictions on people that matched and after the fibers apply the normalized as geodesic fell but the news the slower and slower speed and fundamental spilled 0 so you need discipline to choose a couple functions that starts half year and Europe becomes the wall guests included in this year's becomes clearer and we find something like that to towel off peaks be basically C to wall all of the absolute value of Phoenix the law what most analysts were otherwise otherwise it will exist in which most thank you again that relentless pursuit was going to be played in huge volumes and topology was the generalizes it was discovered by I don't know my side probably the problem was known to have logged in London mentioned times already and in the generalizes the classically there is no doubt that dentist probably invented by then well according to a lot the lights went on in case anyway I want to extend the ban respectively as the lots with normal all 4 Livshits repression but before I really going to miss the bolivar lectured the ratio this was critical .period and it is there a way off this you and so this means that
18:00
away from you I can identify with on and banking here 11 runs and keep the and came the fiber and I'm moving in my violent from so what I'm going to get here is exactly s of gave well as assistant vanishing management fear the singularity is usually 1 critical point going with define upwards of hope in the I'm talking so that's kind of information OK so know we can continue in Italy and London formalism sold Wonderland heightened awareness the president would those have found that out on the Agenda went on here at the you always have to have the ranching and when fix 101 was very few think someone should I think it might become on the link but if you fix just 1 so that's the of wellknown factor which is explained I wouldn't want to differently manifold is a Lagrangian sphere there has been the single eliciting Gration having this month for the fiber In this owners feel vanishing sphere of some parts of India the II you basically have to take the mapping also that the kind of Fiat yes the Mexicans were OK so that wasn't good moments so what so fix pain Livshits fibration being by the way I I want to insist I will not allow also necessary the bridges without singularly it's just products so yeah it was time for and please allow so that's vessels already what I want to include this so what is tonight and go but
20:35
is also the not and I'm not going to write knowledge by medical borders In the is the 1st of the landmines and submit for the unique is a planned under the many forms of the game in the this so that sits in the electrical Buddhism inside the EU such that this was
21:13
less real this doesn't have In a few cases continued to hear sold what that once was the so far as the model when you restrict Of the projection To me so that have from people seem like this of and I II and III will identify seems all too many times in this way will commence appropriate the robbery and there exists the huge room number for which the following :colon when you take the inverse image 12 minus infinity they I find out so that's acceptable then this looks the nicely this looks like the union of race I'll probably so there's erased Q With implied violence London's in the fiber With a fuel in London and let it stand in the wake of pictures of and the NJ the just you are a raise so is the picture so here is coordinate White Xerox said in here is unique because out so that's how the
23:18
rejection of the but when of the looks like sobering at some something the shortest no hole structure some when from the settlement that song the middle class this is really and what really too but this rate is up so basically the looks as follows it is a phenomenon Latin submanifolds inside and at the infinity minus infinity it is complete intricate so it looks like a rate horizontal rain tires but many followers 174 get and the array of Banks of manifolds and don't want to be in OK but this looks and here is like everything is being sold is my substitute you can do this you can make modifications without the pain conditions there is no problem you just need to insist that these things of the you environment about transport but there was no that into a lot of keeping well aware of land management captain and wanted to get them out of the right of the year different fibers and played in the final this is why the condition as it is very convenient no no no no I again it's not the food from the Commission you also to assume that it's proper and I want proper mental was it was that nothing goes to infinity in this direction the doors of projected execs as students proper and use this laughing the IOC member but at the end of the definitions become known and as the thing was going that the year that doesn't yet a definition not fully 1 down with life OK so yet not overly right yeah yeah this my equipment would yeah I'm going the exactly because of where I live you could say that the landlord of sitting here together you know that it's not aboard the visit this many mounting them so on science not reliable monitoring unit the trumpet here to be applicable to them in the classical but the University of infinity is very convenient the OK well not aggressive or I'm going until someone loyal feeling so I need to restrict the Bible Leininger I'm playing in so without you know think you're watching In many ways dedicated to ideas are a source code is the deletion coming up with what kind of problems you get when you the girls who on the different settings of I What I want adopt the most conservative the setting of glass of land wine and this is only a full analysis log I think that from Margarite point of view of this it's not so important that it be so I would assume that there's even 1 McDonald's all of that so exempted already interested and uncle but rights on Monday .period Council again without going into a very low the the missing when have lot for the likelihood of a relative transplants for the muscle thing that which you know it's when to take a twodimensional apology this was bound to on the Lagrangian it measures somehow the complexity of the let the manager hello some bundle inside the syntactic Monday you to abate and these measures OK that's the muscle index but the enamel class the other classes the syntactic area well done and that is the glasses on Web later if 1 is positive then the others positive licenses international made like opposing that model will find so this amusing but then it has positive area it will also have positive lost in this Moslem bandits appears as if willing when better they mention Spitzer's awful Norfolk disks and hopefully 1 is the lessons to be positive and funky along you go into into the topic you will experience this In some right and we will is over we will work with the nautical ranks the article during the foot of the government and now we need to go into the black abilities and once again that's last topic but so I cannot give from definitions but we don't play the following year so we fixed the date clashes with relation to claim the title .period secure and I'm going to assume that the slaughter of fuel is the upper half plane for convenience and I'll help you the fact that they will is gone the appears in 1 of them is the fibers of survival has the right to bit so you know you take all the dungeons of benefits all your class and is the middle formerly Infinity predictably could affect the Soviets Infinity contains information together the 5 so close the last guy that they're growing which is already on will continue is so this is linear translated the which rectifies the problems that the origin of the nozzle capable hands this is in effect
31:02
that it was not wrangling and similarly limits but less classical he itself gives you of forgot that ability to he and also derived from of the sold here 1 should say something so wonderful objects here objects so they wanted to on the way I define it really medical isn't isn't but in not just for the dungeons and I'm going to really assumed that they all lie about in the upper half with drive from inside of a late that's an element of convenience of broken bones so you want to make a few only as such all such redundancy and is already gives you some kind of problems it's already has problems the problem is that and felt feeling of unity the land to make them the miners intersect transversely before you start counting the all make someone Estonian perturbations in so well you get meet here ourselves but just don't what would and that's 1 over these and they have another 1 and it could mean it could be that at the they just sit there the Rays display inside all you have the race for inside okayed it could be the fibers that and essential point this intersection .period gets multiplied by the ring so if you want to make them transversely acting you need to apply a perturbation which is not competent support OK it is unlikely that problems but before any of the problems you have another problem is that the space miserable perturbations amusing little analysis eventually disconnected and if you bird 1 direction in different directions you get different results so what is actually in the flow complex of saved the BMW so he is 1 his lead was the raised here In the here is nominal I always protect them and visualize them like that so here's W. C In it was the end news flow and I wanted to use this space so that you .period it the 3rd don't say nothing in such a way that now they look like these I need you so that left unchanged In years W keeps Simon available donation that it like and the rest is as before if you need perturbing the but vowed to go ahead and do it that's not the problem here comes the essential Britain so you decide the 2nd Rongen was reported to be cylindrical belong also vocal about and if you do it in 1 way or the other the different result so we are going to do it and the choices in this Joyce has arisen you won't see f off double the vehicle of the community and that noon after all we invite you yesterday here are not of so massive that involving 2 wrong this is not very anonymous and when these matters would say I greatest was appointed wrapped was I'm not interrupt situations at infinity I don't have a little complexity it's more flattered to it's an interministerial different From area this is what they do I forgot to say 1 thing about the Of course if you are going to at always the language of the new need it's going on the way to the end hangs on specified y coordinates never want to mention that now ,comma I'm considering like this the and 1 is hanging on my coordinates once Rainier's tool is hanging on like coordinates to cancel while allowing only and this is an enemy being of big restriction because you can always go with them and put them away another thing that Bucharest that has a question using the land the margins hanging on the race here that they're are not really in this joint inside the manifold in they looked disjointed in the but in the end it could actually be the same sentiment all the other so that's not really to that there is no such as assumptions focus OK so now we can continued Tuesday is all there is the socalled inclusion function so was linked to the front door if you take a crack down on In the plane with the wheel horizontal and passing from the scene that is sold here is a normal phenomenon and here's another 1 so you of such a thing you can you can the problem is that you can basically use it to go from the compatibility of the fiber to the credibility of the feet and
38:51
the this is the only reason right here I the yes the the sold lands should be a knowledge of the like the left but anyway you get I don't that's found all forms of violence as a tool for the people of the on objects it does just before leaving big area I would like to the right and times normal but they cannot because I'm in elections fibration fly right now like these are are what we mean by this well I'm going on the hook can be very useful tools of so I am taking my end here and moving into my blood was it's here In my apologize for that here and getting a lot of bewildered OK the fact that I'm staying it's famous makes it easy to define OK so this is Tracy trade and on the death of her report on the political point yet the critical millions of otherwise was you otherwise it you can literally defines it yeah I have so far has not because he wanted to you and the thing to be bed but sometimes you can make it was because of the the union I know that that's so basically you can you go from here here here here and here you started skating and you end up with the novel that's what's on London's inefficient OK it's an entree effect with a that's ancient defunct so this is evidenced by the mission and objects within reach the the respect the ancient structure and it is the well when find out work of 2 was almost the standard about some of the things that don't want this to the answers but I have the honor of the health and that this is not an answer relatively leaves mental None of that going to be a cool so that's going on in my life and I'm going to get that into the yes there is a long way independent for such a valuable that OK alone they so I want to get to the main result of Holland I'm trying to sold here is I'm going to settle for curves with who played this song will not take the curve now I so this is why it was and this is way it was I in excitement scored 2 wins 1 of and that's not why I use discovers and I will use so that these will be used a lot mentally ill also something with so for the most to specify areas so here's my list ratio and here are the critical points it's 1 external the Q as a critical values the values and I think this could these groups so they all end on white because 1 of them In the famous all of them because it's a singular London family and I'm going to call them the 1 deep to undo the key because of England and this thing consolidation midnight organisms into the object of global 3 to 1 in the paper Of all objects while look at the Americans the design I feel like I have arrived at the if the demise of the up and up they here you what they're losing more than they know what they just to the left yeah but I haven't done anything was the 1 with the new there was I going to lose them something OK so now I can stay the is so basically what is the question I was 1 would like to know is so when you have to do so the program capital the Indian this inclusion in front of the How larger which extends for Quayle's knowledge of the glow em In the surprising Council not much larger on on the delivery of care but in the last 4 you the following things for wearing goggles plan alone and you have the following his office and the size of conflict the ride like this call so this is nothing to do with the employee combining the improvization of all of medics I spent for iterated going and it's Michael I'll explain what they mean they directed the entertained fans the 1 that all products with some vector space won the court this is a role that was some political space thinks medical space and then want to the kid this is the beginning cannot knock on wood was asked
46:34
the lead I was take notice of the Isle of Lewis won only once in surprisingly good 1 so here is the that's and 1 2 so that's very critical the capital is the families and will be while country
47:18
came out just of London musical so little spaces of course they depend on the button the canonical only the Grand 2 the gunmen so by calling in the following start from here the is that you have a map to take call and you another elected technical so that the abuse of notation to write but looking this year IBM here's gives it is closed yes yeah that's right OK so 1 of the really say that that you know every on the behind them so hard to think of the highlevel if you're anyone defense have from 1 of these at the end all you need to know on the ends of the dungeons and some mental spaces associated with the only not not too the battles being the flow so and the family the 2 businesses have yourself said he would get out of the elections being interconnected the no OK now ,comma the following thinks so now gun not years yes yes for a finite the now ,comma the following think you'll many wanted to 2 of the following questions organic emboldened medical Buddhism uses some the composition inside any of the Lord and the end of the day want to study and 2 he writes so you may think of user just as the auxiliary I think the main object of study but this 1 some of the possible to doors and to expand overseas another dentist will appear by the way you can find resigned was also when you don't have similar to the board and this is what makes the world premieres but was the I needed so if you don't have seen is appalled that all of these things they good and that is just saying that the isn't iteration ,comma of this so that's what also interest OK so I to focus so the composition the so we can find 1 of the following things for life and in my community and my critical report of the sky and I will think based .period when we think is this past In this In this past week use for me here or in the fight over here the many land on John finishing the engines fears so danish Inc spheres it would be less while the best as he could and would use them and let me know he will be annulled the bulldozers the use of the In the delightful guy 10 yeah the following the composition sir I was really was that the lowest 1 squad has all the tools needed to make the call now comes the most skillful so you should take the banned list inverse order last longer Peter and composers of the last 2 years composing is this 1 :colon people opposing the move to well the on this 1 there is a reason you want you touch these
53:22
things to know about the same no you New :colon escape creative use of the reminds 1 of 1st but you stop once before I felt this week reversed 1 is full of cancer was installed and you continue like this and the other 2 gets too In this case the years for this killing minds want the world to just so there from within the Caymans Swan no Will this case could be here now the more civilized but which is an hour ominous won't be that so yeah it's a long for another what can you do when you get hospital I am aware I think they will satisfied with the answers and you have they have to do is available this summer but this is what what what happens right so you have to take enlarges it's that appeared here a Dutch name 1 to the other big success costs and then the vanishing spheres and continue watching but there's only so much of this it seems so anybody when you don't have any similarities the results is that the the dungeon and do the then 1 here isn't directed ,comma over all the rest 2 booking so I think I have whole much I think so that choices is a goal in the last year and incomprehensible to new 4 minutes the Fireman's Fund wall I want to motivate why do I have to be here so there are many answers Noble of them all along with convincing 1 answer is that 1 plant in 1 year this up you know that's a little too so now Of course you would say the following things that you see on the life of backup only which is kind of notoriously underwriter I think so you know you take languages of many and then adds to the woes of the extreme underwrite bonds it's like mapping :colon off mapping golds of making calls and so on and then in the end you get something that underwriting behaves better about but you had that some very weird objects that what is the relation between these objects and dormitories only borrowed sold this approach that eventually that many many times the cause students the election by something dormitory so if you want to get chorus of interesting :colon assume they can produce interesting Awad isn't this what you want and gone because we accepting cancers of the viscerally use logic but I want to so that's that's 1 motivation to study using tells you a lot of you have year only 1 generation where his theories that the report so the question is you are asking known do which extends from the modern right you get the global businesses the deal the other set of faces containing how large abstract models was coming from you know and I salute them with at all we don't know I wouldn't expect that all models we've got not even in the triangulated the but at the same time we all playing with some approach that takes also into account matters like I was in school designs and then only answer seems to be yes you get everything in this world this would be a complete John legislation of good Defalco but good will that but I don't remember much more some of the other divisions standings coming from referred redolent of platinum brief here that they want to extend the principle soap 1 thing but when you look at bright 1 of of the things that you see is that if you take a complex many flaws it sometimes happens that it can be handled with completely different words think so some of most famous examples of providing all the literary quality and so when it can take say and he was CP was forced to wipe and here are 2 different foods structure is 1 of the 1st "quotation mark Norfolk pollution was just saying that 0 the call that uses the only 1 that was Tuesday 0 but isn't 1 of whom the new 0 well 1 in the real reason but he was just in this case is that of the 1 across of the please but the book but the amount of resources which is the fulllength CEA 1 In the old ways of using that was kind of just still you switched the nights and then went the user about them 1 of the 2 0 1 0 1 In the I really look forward please the nonsense so it's going go kind of the question is OK so we you again have many restructure What is the relation between them now in general I don't know how to answer such questions in general but under some situations the extreme you can actually claims that's different for the structures of Wimbledon the magical inside election British will here however also if links inside simply and knowledge and it is projected projected no I don't have to call it's itself a projected really complex many people so it's a complex manifold and use real structure to real complex conflicts in France then you know you can they cut open sections of it and you can buy implies a space of viable sections where voice opinions look at the diversity and each point here is little reliable Planagan got links with disciple soldiers who thinks and you can cut it was a high publicity so that the people on the streets and if you have a point you can look at it 6 intersecting Bruce with page flounder that's into London and you know you're in on the side of now if you are not lacking and this this age under the extending transversely the single front and that's actually how you create finishes tribulations so human side we have the discretion locals call the level I think it's staff basically what happens is that the compliment all of this here is of complex for the mention of this once and for all the common disconnected this means that all this and around as they are buying was argument and at the there will be a no more fickle President however if you know get the real Lucy I vote things biomedicalimaging age so but if Hicks is embedded in a real way you can consider ring hyperbole in sections so heavily sections of the environment and the reinvention of peaks I think we that this is not the time in yes I am aware of their this land and interests received no restraints Maryland keeps keepsakes and so I think smiled so there is an interesting way to his work things can only get taken green and blue like much the things like that that's equivalent so focus on just 2 finish story salt you can only get the display and locals where things and this is size of the all with CBN was really no this is this comic so here you have this Albion and they decided to go many chambers many many chambers of instructors and this is exactly what it was you so if you take so here is for example the restructure the using of pure uncles of B1 and here the 1 give a story In now exactly what was exactly but if you you say when you look at the problem sections and you take a pencil assault with pencil inside the recipient the hits the ball belied the transversely you get but less expensive you people the base not most new that definitions the duration and if you take the bounds of the you going to get precisely the Board of Elections who relations and inside of the London borders we not reliable borders and that that's not the problem was 1 Starting here and the other G so Saudi women engineers the as a global that he was projection is going to look like a the is going fool the singularity is that there really so some the singularities only some of them are you going to do it .period so each time you hit the water you worked through civility and here you have a real but was planning to this period real boggles funding to and if you want the world to be in the denial of this on the and was banned so Ramiscal dozens of their peers from really was flat just and I cannot say any results but that's what the hope is that using this this the compositions you could say something about the relation between the topology of different ends thousands of all starts with critics and critical values that I think of the year know this is somewhat of a large numbers is much lower and the thing is the whole thing is what year's it's a small buildings I mean that's almost none of them but it's not how to just because you say that you get back to the elections fibrillation and incited to get real mission which becomes was 1 of the 2 of you know yes yes if so of course you OK so that's a distance the from nature of I really the best thing we can do with this problem at "quotation mark settlement In view of the conflict of the the you can call it that society you have to my shame I about his references enough I don't know if there's a relations the the we expect the what would From this point of the Rangers and later as to the use of not without huge restrictions on what you want and you can the potential I think think not everything is this week will appear as a real about and in the Belisle classical topological restrictions on such things as but what is benefiting more interesting is the same way as that organs such pictures what you get many times is the real cost is a Japan connected even on the backs surfaces ring once you get the real costs can be for copies of a sphere appeals to other components and then in other commodities which is very special Russell the logic here is really is joining the union of 1 and maybe again confront restrictions on this take it from citizens of the Union fine the fact of the matter I you I would the 420 the which In this gesture of I would some the president of the National Election there was no the 1st year well obviously but that that is the 1 1 result which good no but it had nothing to do do with me because the effect on the tour was being Lagrangian so this goes back to the devil so to the government that refund on influence dimensions really mentioned 6 or more showing that the real about can never be hyperbole but then later on I think I think whether it's a series of the national and so inevitable given the hopeful extended 2 1 at In such things so there the fact that some ended up the doubles prove effective it's really played a role in the world future but you know that this is what I am and that's a serious restrictions I I'm aware of many others the downloadable their nonconference opponent what is that of the University of it was the 2nd some money it's in the book no I think I can answer that part of the game so if you if you will it if you opt not to find funding if you will find many fund ,comma have lot what's important I really think it is consuming more and this is this is how long the approval the performance of its final and the young it then close runs a metaphor there inside it it cannot have you can negatively the fact that this is yeah I would define it yet but I forgot the future if it's in funeral view of this summer something the gentle us to look at up that is satisfied in the final he added the balance of a united given them in the province basically the following you take this spheres we have to fix fares if you look at the little neighborhood of random you stretch the next in the list of the of the game stretch Mac and I have had some kind system building right so well you will not even get to sewing the abandonment of the land manager and that's where you stretch the thing that you wanted geodesic 6 you know that the thing is hyperbolic all of these issues ahead hyperbolic rhetoric than you could calculate the index of this thing and get them contradictions 2 of them index that you come from the fund right now the profits of something this idea and downloading something quite different with a graphical mortgages